Let $M$ be a smooth compact manifold of dimension $n$, and let $U$ be a smooth compact manifold with boundary, of the same dimension $n$, embedded in $M$.

The embedding induces maps on $\pi_1$.

If $\pi_1(\partial U) \to \pi_1(M)$ is injective, does this imply that $\pi_1(U) \to \pi_1(M)$ is injective?

If true, can you direct me to a reference or a short proof?

EDIT: I reformulated the question adding compactness of M and U to rule out the counterexample given in an answer.